Higher Engineering Mathematics, Sixth Edition

616 Higher Engineering Mathematics

3. For the waveform shown in Fig. 66.6 deter-
   mine (a) the Fourier series for the functionand
   (b) the sum of the Fourier series at the points
   of discontinuity.
   ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
   (a)f(x)=

1

2

+

2

π

(

cosx−

1

3

cos3x

+

1

5

cos5x−···

)

(b)

1

2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ x

f(x)

0

1

2  3 

2

23  

2

2 

2



2

Figure 66.6

4. For Problem 3, draw graphs of the first three
   partial sums of the Fourier series and show that
   as the series is added togetherterm by term the
   result approximates more and more closely to
   the function it represents.


5. Find the term representing the third harmonic
   fortheperiodicfunctionofperiod2πgivenby:

f(x)=

{

0 , when −π<x< 0

1 , when 0<x<π

[

2

3 π

sin3x

]

6. Determine the Fourier series for the periodic
   function of period 2πdefined by:

f(t)=

⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

0 , when −π<t< 0

1 , when 0<t<

π

2

− 1 , when

π

2

<t<π

The function has a period of 2π.

⎡

⎢⎢

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

f(t)=

2

π

⎛

⎜⎜

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

cost−

1

3

cos3t

+

1

5

cos5t−···

+sin2t+

1

3

sin6t

+

1

5

sin10t+···

⎞

⎟⎟

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎤

⎥⎥

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

7. Show that the Fourier series for the periodic
   function of period 2πdefined by

f(θ )=

{

0 , when −π<θ< 0

sinθ, when 0<θ<π

is given by:

f(θ )=

2

π

(

1

2

−

cos2θ

( 3 )

−

cos4θ

( 3 )( 5 )

−

cos6θ

( 5 )( 7 )

−···

)